Natural orbital eigenvalue equation

In recent years density-functional methods32 have made it possible to obtain orbitals that mimic correlated natural orbitals directly from one-electron eigenvalue equations such as Eq. (1.13a), bypassing the calculation of multi-configurational MP or Cl wavefunctions. These methods are based on a modified Kohn-Sham33 form (Tks) of the one-electron effective Hamiltonian in Eq. (1.13a), differing from the HF operator (1.13b) by inclusion of a correlation potential (as well as other possible modifications of (Fee(av))-... 

The molecular orbital coefficients (/, /i = 1,2, , m) which specify the nature, and hence, energy of the orbital ///, arc determined by solving the eigenvalue equation of the effective one-electron Hamiltonian,associated with the molecule (equation 1.11) ... 

A matrix is positive semidefinite if and only if all its eigenvalues are nonnegative. Equation 1.19 (or the corresponding bounds on the natural orbital occupation numbers, Equation 1.17) is the A-representability constraint for the one-matrix. That is, every one-matrix that satisfies Equation 1.19 can be derived from an average of A-electron states, as in Equation 1.11. 

Another insight into the nature of a covalent bond is provided by analysing the anisotropy of the electron density distribution p (r) at the bond critical point p. For the CC double bond, the electron density extends more into space in the direction of the n orbitals than perpendicular to them. This is reflected by the eigenvalues 2, and k2 of the Hessian matrix, which give the curvatures of p (r) perpendicular to the bond axis. The ratio 2, to /.2 has been used to define the bond ellipticity e according to equation 8S0 ... 

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